Coupled Sequences of Generalized Fibonacci Trees and Unequal Costs Coding Problems

Fibonacci trees and generalized Fibonacci trees have been defined and studied by Horibe [4], Chang [2], and the author [1]. The k^ tree in the sequence of r-ary generalized Fibonacci trees, T(k), has T(k-c(i)) as the I leftmost subtree descending from its root node for k>r, and T(k) consists of a single root node for k-\ ...,r. Here c(i), z' = l,...,r, are positive integers with greatest common divisor 1 and are nondecreasing in z. In the case in which r = 2, c(l) = 1, c(2) = 2, the generalized Fibonacci trees are the Fibonacci trees of Horibe [4]. In addition to the construction of generalized Fibonacci trees by the recursive specification of their subtrees, there is an equivalent construction by the method of "types." The two constructions can be seen to be equivalent by induction. In the method of types, each leaf node is assigned one of c(r) "types" denoted by aha2, ...,ac(r). Then T(k + l) is constructed from T(k) according to the following set of rules. A leaf node of type ax in T(k) will be replaced by r descendant nodes of types ac(1), ac(2),..., ac(r) in left to right order in T{k +1). A leaf node of type Qj in T{k) will be replaced by a node of type aj_x in T(k + l), j = 2,..., c(r). The sequence of trees begins with T(T) which consists of a single root node of type ac^. The construction by leaf node type has an interpretation in connection with Yarn's algorithm for the solution of a particular unequal costs coding problem [5]. Thus, the recursive subtree method also generates Yarn's code trees. In the coding problem, a path from the root to a leaf describes a codeword, a sequence of r-ary symbols used to represent the source symbol assigned to the leaf. It is assumed that the code trees are exhaustive, that is, every interior node has exactly r descendants. In the case that the I code symbol, i = 1, ...,r, costs c{i), the generalized Fibonacci tree minimizes the average codeword cost for equally likely source symbols when the number of leaives in the generalized Fibonacci tree is the same as the number of source symbols. In Yarn's algorithm for the optimal code tree, leaf nodes of least cost, say c, in an optimal tree for a given number of leaves are replaced by r descendant leaf nodes of cost c + c(i), i 1,..., r, in left to right order in generating the optimal tree for the appropriate larger number of leaves; the sequence begins with a single root node of cost 0. The correspondence between Yarn's algorithm and the construction of generalized Fibonacci trees by the method of types is immediate because the method of types is exactly a mechanism for keeping track of each leaf node until it is a node of least cost in Yarn's sense. Easy recurrence relations for the resulting number of leaves and average code cost can be obtained through the recursive subtrees perspective. In this paper, a further generalization of Fibonacci trees is examined: the case of multiple coupled, recursively-generated sequences of trees. These sequences of trees have interesting structure and, under certain conditions, can be interpreted as optimal code trees for a generalization of Yam's unequal costs coding problem. One arbitrarily selected example will be considered